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Reports on Mathematical Physics

Application of noise sequences of infinite matrices in the theory of quantum observables

PEKKA J. LAHTI, Maciej Mączyński, Egon Scheffold, Kari Ylinen

Published Aug 1, 2006• 0 citations
Publisher
DOI
10.1016/s0034-4877(06)80043-1

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Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.
We exhibit an operator norm bounded, infinite sequence {An}\{A_n\}{An​} of 4n×4n4n \times 4n4n×4n complex matrices for which the commutator map X↦XAn−AnXX\mapsto XA_n - A_nXX↦XAn​−An​X is uniformly bounded below as an operator over the space of trace-zero self-adjoint matrices equipped with Hilbert--Schmidt norm. The construction is based on families of quantum expanders. We give several potential applications of these matrices to the study of quantum expanders. We formulate several natural conjectures and problems related to such matrices and provide numerical evidence.